A detailed imaging analysis of the globular cluster (GC) system of the Sombrero galaxy (NGC 4594) has been accomplished using a six-image mosaic from the Hubble Space Telescope Advanced Camera for Surveys. The quality of the data is such that contamination by foreground stars and background galaxies is negligible for all but the faintest 5% of the GC luminosity function. This enables the study of an effectively pure sample of 659 GCs until ~2 mag fainter than the turnover magnitude, which occurs at MTOMV=-7.60+/-0.06 for an assumed m-M=29.77. Two GC metallicity subpopulations are easily distinguishable, with the metal-poor subpopulation exhibiting a smaller intrinsic dispersion in color compared to the metal-rich subpopulation. There are three new discoveries. (1) A metal-poor GC color-magnitude trend has been observed. (2) The fact that the metal-rich GCs are ~17% smaller than the metal-poor ones for small projected galactocentric radii (less than ~2') has been confirmed. However, the median half-light radii of the two subpopulations become identical at ~3' from the center. This is most easily explained if the size difference is the result of projection effects. (3) The brightest (MV<-9.0) members of the GC system show a size-magnitude upturn, where the average GC size increases with increasing luminosity. Evidence is presented that supports an intrinsic origin for this feature rather than being a result of accreted dwarf elliptical nuclei. In addition, the metal-rich GCs show a shallower positive size-magnitude trend, similar to what is found in previous studies of young star clusters. Based on observations made with the NASA/ESA Hubble Space Telescope, obtained (from the data archive) at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. These observations are associated with program 9714.